Solution to Problem 8: "Snake in the Grass"

The word "neighbor" used here means in all eight directions; "HV-neighbor" means only the four horizontal or vertical neighbors. Here are a few rules:

(A) If a spot in the path has two HV-neighbors on the path, then they are adjacent on the path and any other HV-neighbors are not on the path. In addition, any diagonal neighbors that are not HV-neighbors to exactly one of the HV-neighbors of the original spot are not on the path.

(B) If a spot is labelled with a 1, then its neighbor on the path must be diagonal.

(C) If a spot on the path has only two HV-neighbors that can be on the path, then they are on the path. If a spot does not have two HV-neighbors that can be on the path, then it cannot be on the path.

snake-A.bmp (162062 bytes) A16 isn't in the path, so B16 can be filled, and the path goes A14-A15-B15-C15-C16 (C14 can be filled) -D16-E16-F16 (E15 is filled) - G16.

D1 can be filled, and the path goes E1-E2-D2-C2-C1-B1-A1-A2-A3-A4. The only neighbor of G7 that can be on the path is H6.
snake-B.bmp (162062 bytes) If the path I6-H6-H5 continued to H4, then the path that went around K2 must leave through the vicinity of K10 by both paths, but there is no way to do this without breaking K10. Therefore, G5 is on the path. Even with this, the only way to fill the area around K2 is to have the path go H6-I6-J6-J5-J4-J3-I3-I2-I1-J1 and so on. J8 must be in the path to fill I7. J10 must be in the path, or else K10 would break. H9 must be in the path to fill I8. F4 must be in the path to fill G3.
snake-C.bmp (162062 bytes) If A5, A6, or A7 were in the path, then B6 would break. A9 can be filled. If B7 were in the path, then B6 would break. The path contains B4-B5-C5-D5-D4. The path must contain L14-L13-L12-K12-J12, so I11 is filled.
snake-D.bmp (162062 bytes) Given that the path cannot cross itself diagonally or contain an incomplete circuit, most of the remainder can be solved, filling in K13 and G14 along the way. F10 must be in the path to fill G11, which gives the rest.
snake-E.bmp (162062 bytes) The solution, representing the number of used path spots in rows 2, 5, 8, 11, and 14 is 6 8 6 1 5.

Comments: I tried to solve this one during the contest (like the other two path puzzles), but did not finish.  I don't believe I appreciated all of the ramifications of the no-loopback rule at the time.